Convex duality in optimal investment under illiquidity

Math. Program., Ser. BDOI 10.1007/s10107-013-0721-5

FULL LENGTH PAPER

Convex duality in optimal investment under illiquidity

Teemu Pennanen

Received: 11 March 2013 / Accepted: 15 October 2013© Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2013

Abstract We study the problem of optimal investment by embedding it in the generalconjugate duality framework of convex analysis. This allows for various extensionsto classical models of liquid markets. In particular, we obtain a dual representationfor the optimum value function in the presence of portfolio constraints and nonlineartrading costs that are encountered e.g. in modern limit order markets. The optimizationproblem is parameterized by a sequence of financial claims. Such a parameterizationis essential in markets without a numeraire asset when pricing swap contracts andother financial products with multiple payout dates. In the special case of perfectlyliquid markets or markets with proportional transaction costs, we recover well-knowndual expressions in terms of martingale measures.

Keywords Optimal investment · Illiquidity · Convex duality

Mathematics Subject Classification 90C25 · 46A20 · 91Gxx

1 Introduction

Convex duality has long been an integral part of mathematical finance. Classical ref-erences include Harrison and Kreps [15], Harrison and Pliska [16], Kreps [25] andDalang et al. [8] where the no-arbitrage principle behind the Black–Scholes formulawas related to the existence of certain dual variables; see Delbaen and Schacher-mayer [10] for a detailed discussion of the topic. In problems of optimal investment,convex duality has become an important tool in the analysis of optimal solutions; seee.g. Kramkov and Schachermayer [23], Delbaen et al. [9], Karatzas and Zitkovic [21]and Rogers [40] and their references.

T. Pennanen (B)Department of Mathematics, King’s College London, London, UKe-mail: [email protected]

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This paper studies convex duality in problems of optimal investment in illiquidfinancial markets where one may encounter frictions or restrictions when transferringwealth through time or between assets. Our model extends the classical linear model byallowing for portfolio constraints and nonlinear transaction costs that are encounterede.g. in limit order markets. In particular, we do not assume, a priori, the existence ofa numeraire that would allow for free transfer of (both positive as well as negativeamounts of) wealth through time. This is a significant departure from the classicalmodel where a numeraire is assumed in order to express wealth processes in terms ofstochastic integrals. Such models are at odds with real markets where much of tradingconsists of exchanging sequences of cash-flows. Such trades would be unnecessary ifone could postpone payments to a single date by shorting the numeraire.

While in classical models of perfectly liquid markets, convex analysis can often bereduced to an application of basic separation theorem, in the presence of nonlinearilliquidity effects, more sophisticated convex analysis is needed; see for example Cvi-tanic and Karatzas [6], Jouini and Kallal [17,18], Guasoni [14], Schachermayer [43],Rokhlin [41], Bielecki, Cialenco and Rodriguez [4] as well as the books of Kabanovand Safarian [19] and Rogers [40]. The present paper builds on the market model intro-duced in Pennanen [30] where trading costs and portfolio constraints are describedby general convex normal integrands and measurable closed convex sets, respectively.This discrete-time model provides a unifying framework for modeling transactioncosts and portfolio constraints as well as nonlinear illiquidity effects.

The classical superhedging principle was extended to this model in [32,33]. Thepresent paper makes similar extensions to the duality theory of optimal investment.More precisely, we derive a dual representation for the optimal value function ofthe optimal investment problem studied in [34] as a basis for valuation of financialcontracts. The dual representation is obtained by applying the stochastic optimizationduality framework developed in [31] as an instance of the conjugate duality frameworkof Rockafellar [38]. This allows for a unified treatment of many well known modelsof mathematical finance where dual correspondences have often been derived case bycase. In particular, we illustrate how the dual variables in the general case are connectedto martingale measures and shadow prices that have been extensively studied in modelswith a numeraire.

2 Optimal investment

This section recalls the optimal investment problem from [34]. The problem is para-meterized by financial liabilities characterized by a sequence of cash-flows the agenthas to deliver over time. The dependence of the optimum value on the liabilities isimportant in valuation of swap contracts and other financial products that provide pay-ments at multiple points in time. Section 3 derives dual expressions for the optimumvalue function.

Consider a financial market where a finite set J of assets can be traded over finite dis-crete time t = 0, . . . , T . We will model uncertainty by a probability space (�,F , P)

and the information by a nondecreasing sequence (Ft )Tt=0 of sub-sigma algebras of

F . At time t one does not know which scenario ω ∈ � will realize but only to which

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element of Ft it belongs to. The filtration property Ft ⊆ Ft+1 means that informationincreases over time.

We describe trading strategies by (Ft )Tt=0-adapted sequences x = (xt )

Tt=0 of

RJ -valued stochastic processes. The random vector xt describes the portfolio of assets

held over (t, t + 1]. The adaptedness means that xt is Ft -measurable, i.e. the portfoliochosen at time t only depends on information available at time t . The linear space ofadapted trading strategies will be denoted by N . Unless FT has only a finite number ofelements with positive probability, N is an infinite-dimensional space. We will assumethat F0 = {�,∅} so that x0 is independent of ω. Implementing a portfolio processx ∈ N requires buying a portfolio �xt := xt − xt−1 of assets at time t . Negativepurchases are interpreted as sales.

Trading costs will be described by an (Ft )Tt=0-adapted sequence S = (St )

Tt=0 of con-

vex normal integrands on RJ ×F . This means that for each t , the set-valued mapping

ω �→ {(x, α) ∈ RJ × R | St (x, ω) ≤ α} is closed convex-valued and Ft -measurable.1

The value of St (x, ω) is interpreted as the cost of buying a portfolio x ∈ RJ at time t

in state ω. Accordingly, we assume that St (0, ω) = 0. Implementing a trading strategyx ∈ N requires investing St (�xt ) units of cash at time t . The measurability conditionon St implies that ω �→ St (�xt (ω), ω) is Ft -measurable, i.e. the cost of a portfoliois known at the time of purchase. Indeed, the cost is the composition of the functionsω �→ (�xt (ω), ω) and St , where the former is (Ft ,B(RJ ) ⊗ Ft )-measurable whenx ∈ N while the latter is B(RJ ) ⊗ Ft -measurable, by [39, Corollary 14.34]. Here, asusual, B(RJ ) stands for the Borel-measurable subsets of R

J .The classical model of perfectly liquid markets corresponds to St (x, ω) = st (ω) · x

where s = (st )t=0 is an (Ft )Tt=0-adapted sequence of price vectors independent of the

traded amounts. Markets with proportional transaction costs and/or bid-ask-spreadscan be modeled with sublinear cost functions

St (x, ω) = sup{s · x | s ∈ [s−t (ω), s+

t (ω)]},

where the (Ft )Tt=0-adapted R

J -valued processes s− and s+ give the bid- and ask-prices, respectively; see [18]. Convex trading costs arise naturally also in modernlimit order markets, where the cost of a “market order” is nonlinear and convex in thetraded amount. Parametric convex market models have been proposed e.g. in Çetinand Rogers [5] and Malo and Pennanen [26].

We model portfolio constraints by requiring that the portfolio xt chosen at time thas to belong to a closed convex set Dt much as e.g. in [11,28,41]. We allow Dt to berandom but assume that it is Ft -measurable. We assume that 0 ∈ Dt almost surely, i.e.that the zero portfolio is feasible. The classical unconstrained model corresponds toD ≡ R

J while short selling constraints, studied e.g. in Cvitanic and Karatzas [6] andJouini and Kallal [17] can be described by D ≡ R

J+. As observed there, short sellingconstraints can also be used to model different interest rates for lending and borrowing.Indeed, this can be done by introducing lending and borrowing accounts whose unit

1 A set-valued mapping T : � ⇒ Rn is Ft -measurable if {ω ∈ � | T (ω) ∩ U �= ∅} ∈ Ft for every open

U ⊂ Rn .

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T. Pennanen

prices appreciate at lending and borrowing rates, respectively, and by restricting theinvestments in these assets to be nonnegative and nonpositive, respectively.

Consider now an agent whose financial position is described by an (Ft )Tt=0-adapted

sequence of cash-flows c = (ct )Tt=0 in the sense that the agent has to deliver a random

amount ct of cash at time t . We allow ct to take both positive and negative valuesso it may describe both endowments as well as liabilities. In particular, −c0 may beinterpreted as an initial endowment while the subsequent payments ct , t = 1, . . . , Tmay be interpreted as the cash-flows associated with financial liabilities. We willdenote the linear space of (Ft )

Tt=0-adapted sequences of cash-flows by

M = {(ct )Tt=0 | ct ∈ L0(�,Ft , P)}.

Here and in what follows, L0(�,Ft , P) stands for the linear space of equivalenceclasses of Ft -measurable real-valued functions. As usual, two measurable functionsare equivalent if they coincide P-almost surely. Given c ∈ M, and nondecreasing2

convex functions Vt on L0(�,Ft , P), we will study the problem

minimizeT∑

t=0

Vt (St (�xt ) + ct ) over x ∈ ND, (ALM)

where ND := {x ∈ N | xt ∈ Dt , t = 0, . . . , T − 1, xT = 0} and �xt := xt − xt−1.We always define x−1 = 0 so the elements of ND describe trading strategies that startand end at liquidated positions. The functions Vt measure the disutility (regret, loss, …)caused by the net expenditure St (�xt )+ct of updating the portfolio and paying out theclaim ct at time t . We allow Vt to be extended real-valued but assume that Vt (0) = 0.We interpret the value of Vt (St (�xt ) + ct ) as +∞ unless St (�xt ) ∈ L0(�,Ft , P).In other words, the formulation of (ALM) includes the implicit constraint that �xt ∈domSt almost surely.

Problem (ALM) can be interpreted as an asset-liability management problem whereone looks for a trading strategy whose proceeds fit the liabilities c ∈ M optimallyas measured by the “disutility functions” Vt . Despite the simple appearance, (ALM)covers many more familiar instances of portfolio optimization problems. In particular,when3 Vt = δL0− for t < T , we can write it as

minimize VT (ST (�xT ) + cT ) over x ∈ ND

subject to St (�xt ) + ct ≤ 0, t = 0, . . . , T − 1. (1)

This is an illiquid version of the classical utility maximization problem.

2 A function Vt : L0(�,Ft , P) → R is nondecreasing if Vt (c1t ) ≤ Vt (c2

t ) whenever c1t ≤ c2

t almostsurely.3 Here and in what follows, δC denotes the indicator function of a set C : δC (x) equals 0 or +∞ dependingon whether x ∈ C or not.

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Example 1 (Numeraire and stochastic integration) Assume, as e.g. in Çetin andRogers [5] and Czichowsky et al. [7], that there is a perfectly liquid asset (numeraire),say 0 ∈ J , such that

St (x) = x0 + St (x) and Dt = R × Dt ,

where S and D are the cost process and the constraints for the remaining risky assetsJ = J \{0}. We can then use the budget constraint in (1) to substitute out the numerairefrom the problem. Indeed, defining

x0t = x0

t−1 − St (�xt ) − ct t = 0, . . . , T − 1,

the budget constraint holds as an equality for t = 1, . . . , T − 1 and

x0T −1 = −

T −1∑

t=0

St (�xt ) −T −1∑

t=0

ct .

Substituting this in the objective (and recalling that xT = 0 for x ∈ ND), problem (1)becomes

minimize VT

(T∑

t=0

St (�xt ) +T∑

t=0

ct

)over x ∈ ND,

which shows that in the presence of a numeraire, the timing of payments is irrelevant.This is the problem studied in [5] in the case of strictly convex St and in [7] in the caseof sublinear St . In linear market models with St (x, ω) = st (ω) · xt , we can expressthe accumulated trading costs as a stochastic integral,

T∑

t=0

St (�xt ) =T∑

t=0

st · �xt = −T −1∑

t=0

xt · �st+1.

We then recover constrained discrete-time versions of the utility maximization prob-lems studied e.g. in [2,3,9,23,24]. In [2,23,24], the financial position of the agent wasdescribed solely in terms of an initial endowment w ∈ R without future liabilities.This corresponds to c0 = −w and ct = 0 for t > 0.

We will denote the optimal value of problem (ALM) by

ϕ(c) := infx∈ND

T∑

t=0

Vt (St (�xt ) + ct ).

It is easily verified that ϕ is a convex function on M. The value function ϕ has a centralrole in the valuation of financial contracts with multiple payout dates; see [34]. In thecompletely risk averse case where Vt = δL0− for all t = 0, . . . , T , we get ϕ = δC , where

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T. Pennanen

C = {c ∈ M | ∃x ∈ ND : St (�xt ) + ct ≤ 0 ∀t}is the set of claim processes that can be superhedged without a cost; see [33] for furtherdiscussion and references on superhedging. On the other hand, since the functions Vt

are nondecreasing, we can write ϕ as the infimal convolution

ϕ(c) = infd∈M

{T∑

t=0

Vt (dt )

∣∣∣∣∣ c − d ∈ C}

(2)

= infd∈C

T∑

t=0

Vt (ct − dt ),

of δC and the function

V(d) =T∑

t=0

Vt (dt )

as is seen by first writing (ALM) in the form

minimizeT∑

t=0

Vt (dt ) over x ∈ N , d ∈ M (3)

subject to St (�xt ) + ct ≤ dt , xt ∈ Dt .

The variable d may be interpreted as investments the agent makes to his portfolio overtime. Alternatively, one may interpret −c and −d as endowment and consumption,respectively, in the spirit of the optimal consumption-investment problem of Karatzasand Zitkovic [21] in liquid markets in continuous time.

3 Duality

Prices of financial products are often expressed in terms of dual variables of one kindor another. Prices of bonds can be expressed in terms of zero curves, which representthe time-value of money. In models with a cash-account, on the other hand, prices ofrandom cash-flows are often expressed in terms of martingale measures. In illiquidmarkets without a cash-account, one needs more general dual variables that encompassboth the time value of money as well as the randomness. This section derives a dualrepresentation for the value function ϕ of (ALM) in terms of such variables. Whenspecialized to more traditional market models, we recover (discrete-time versionsof) some well-known duality results for portfolio optimization. Pricing formulas forcontingent claims can then be derived from the general relationships between theoptimal investment and contingent claim valuation; see [34].

Our strategy is to embed (ALM) in the general stochastic optimization dualityframework of [31] which is essentially an instance of the classical conjugate dualityframework of Rockafellar [38]. The bilinear form

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〈c, y〉 := ET∑

t=0

ct yt

puts the space

M1 := {(ct )Tt=0 | ct ∈ L1(�,Ft , P)}

of integrable sequences of cash-flows in separating duality with the space

M∞ := {(yt )Tt=0 | yt ∈ L∞(�,Ft , P)}

of essentially bounded adapted processes. Given a convex function f on M1, itsconjugate on M∞ is defined by

f ∗(y) = supc∈M1

{〈c, y〉 − f (c)}.

Being the pointwise supremum of continuous linear functions, f ∗ is convex and lowersemicontinuous with respect to the weak topology σ(M∞,M1). The classical bicon-jugate theorem says that if f is proper and lower semicontinuous with respect to theL1-norm, then it has the dual representation

f (c) = supy∈M∞

{〈c, y〉 − f ∗(y)}; (4)

see Moreau [27]. This abstract formula is behind many fundamental duality results inmathematical finance. In order to apply it to (ALM), we will first establish the lowersemicontinuity of the value function ϕ.

We will assume from now on that

Vt (ct ) = Evt (ct ) :=∫

vt (ct (ω), ω)d P(ω),

where vt is an Ft -measurable normal integrand on R × � such that vt (·, ω) is proper,nondecreasing and convex with vt (0, ω) = 0 for every ω ∈ �. As usual, we definethe integral of a measurable function as +∞ unless the positive part of the function isintegrable.

Given a market model (S, D), we obtain another market model (S∞, D∞) bydefining S∞

t (·, ω) and D∞t (ω) pointwise as the recession function and recession cone

of St (·, ω) and Dt (ω), respectively. When S is sublinear and D is conical, we simplyhave (S, D) = (S∞, D∞). By [37, Corollary 8.3.2 and Theorem 8.5],

S∞t (x, ω) = sup

α>0

St (αx, ω)

α,

D∞t (ω) =

⋂

α>0

αDt (ω).

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T. Pennanen

The required measurability properties hold by [39, Exercises 14.54 and 14.21]while the convexity and topological properties come directly from the definitions.The following is derived in [34] from a more general result of [35] on stochasticoptimization.

Theorem 2 Assume that {x ∈ ND∞ | S∞t (�xt ) ≤ 0} is a linear space and that for

every t the functions vt (·, ω) are almost surely nonconstant with vt ≥ m for some inte-grable function m ∈ L1. Then the value function ϕ is a proper lower semicontinuousconvex function on M1 and the infimum in (ALM) is attained for every c ∈ M1.

The linearity condition in Theorem 2 is a generalization of the no-arbitrage con-dition in classical perfectly liquid markets; see [32, Section 4]. In particular, whenSt (x) = st · x and Dt ≡ R

J , the linearity condition means that any x ∈ ND withst · �x ≤ 0 almost surely for all t has st · �x = 0 almost surely for all t , i.e.there are no self-financing trading strategies that generate nonzero revenue. This isexactly the no-arbitrage condition. In nonlinear market models, the linearity conditionis implied by the so called “robust no-arbitrage” condition; see [32, Section 4]. In themost risk averse case where vt = δR− for every t , Theorem 2 says that the set C isclosed. In the classical linear model of Example 1, we thus recover the key closed-ness result of Schachermayer [42, Lemma 2.1]. The linearity condition holds also ifD∞

t (ω) ∩ {x ∈ RJ | S∞

t (x, ω) ≤ 0} = {0} almost surely for every t . Indeed, sincex−1 = 0, by definition, this condition implies {x ∈ ND∞ | S∞

t (�xt ) ≤ 0} = {0}. Thiscertainly holds if D∞

t ⊆ RJ+ and {x ∈ R

J | S∞t (x, ω) ≤ 0}∩R

J+ = {0}. Here, the firstcondition means that infinite short positions are prohibited while the second meansthat there are no completely worthless assets.

Nonconstancy of vt (·, ω) in Theorem 2 simply means that the investor alwaysprefers more money to less. The lower bound is a more significant restriction since itexcludes e.g. the logarithmic utility. In Kramkov and Schachermayer [23] and Rasonyiand Stettner [36] such bounds were avoided but, at present, it is unclear if the lowerbound can be relaxed in illiquid markets. However, Guasoni [13, Theorem 5.2] givesthe existence of optimal solutions in a continuous time model with a cash-account andproportional transaction with similar conditions on the utility function as in [23] and[36]. Much like in [36] and the proof of Theorem 2, his approach was based on the“direct method” rather than duality arguments as e.g. in [23].

While Theorem 2 establishes the validity of the dual representation for the valuefunction ϕ, its conjugate ϕ∗ can be expressed in terms of the support function

σC(y) = supc∈M1

{〈c, y〉 | c ∈ C}

of C and the conjugates

v∗t (y, ω) = sup

c∈R

{cy − vt (c, ω)}

of the disutility functions vt as follows.

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Lemma 3 The conjugate of ϕ can be expressed as

ϕ∗(y) = σC(y) + ET∑

t=0

v∗t (yt ).

Proof Problem (ALM) can be written as

minimize E f (x, d, c) over x ∈ N , d ∈ M,

where f : (RJ )T +1 × RT +1 × R

T +1 × � → R is defined

f (x, d, c, ω) ={∑T

t=0 vt (dt , ω) if St (�xt , ω) + ct ≤ dt , xt ∈ Dt (ω), xT = 0,

+∞ otherwise.

Since vt and St are normal integrands and Dt are measurable, it follows from Theo-rem 14.36 and Proposition 14.44 of [39] that f is a normal integrand. We are thus inthe general stochastic optimization framework of [31] so, by [31, Theorem 2.2],

− ϕ∗(y) = infx∈N ,d∈M

El(x, d, y), (5)

where l(x, d, y, ω) = infc∈RT +1{ f (x, d, c, ω) − ∑Tt=0 ct yt }. We can write l as

l(x, d, y, ω) =

⎧⎪⎨

⎪⎩

+∞ unless x ∈ X (ω),∑Tt=0[vt (dt , ω) + yt St (�xt , ω) − dt yt ] if x ∈ X (ω) and y ≥ 0,

−∞ otherwise,

where X (ω) = {x ∈ (RJ )T +1 | �xt ∈ domSt (·, ω), xt∈Dt (ω) ∀t}. Unless y ≥ 0almost surely, the infimum in (5) equals −∞ (it is attained e.g. by (x, d) = (0, 0)).For y ≥ 0,

−ϕ∗(y) = infx∈N ,d∈M

{E

T∑

t=0

[yt St (�xt ) + vt (dt ) − dt yt ]

∣∣∣∣∣ x ∈ X

}

= infx∈N

{E

T∑

t=0

yt St (�xt )

∣∣∣∣∣ x ∈ X

}+ inf

d∈ME

T∑

t=0

[vt (dt ) − dt yt ]

= infx∈N

{E

T∑

t=0

yt St (�xt )

∣∣∣∣∣ x ∈ X

}− E

T∑

t=0

v∗t (yt ),

where the second equality holds since we can restrict the minimization, without affect-ing the infimum, to those x ∈ N and d ∈ M for which both integrands have integrable

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T. Pennanen

positive parts. The last equality comes from the interchange rule for normal integrands;see e.g. [39, Theorem 14.60]. A similar argument in the case vt ≡ δR− , shows that

−σC(y) = infx∈N

{E

T∑

t=0

yt St (�xt )

∣∣∣∣∣ x ∈ X

}

for y ≥ 0 while σC(y) = ∞ for y � 0. ��The form of the conjugate in Lemma 3 is not surprising given the expression (2) of

ϕ as the infimal convolution of δC and V . However, the infimal convolution is takenin the space M of all adapted claim processes, not over M1. One could, of course,concentrate on integrable claim processes from the beginning and study the function

ϕ(c) = infd∈M1

{E

T∑

t=0

vt (dt )

∣∣∣∣∣ c − d ∈ C}

which has the same conjugate and essentially the same economic interpretation as ϕ.However, the proof of Theorem 2 breaks down if d ∈ M is restricted to M1 and wedo not know whether ϕ is lower semicontinuous under the assumptions of Theorem 2.

Combining Theorem 2 and Lemma 3 with the biconjugate theorem gives the fol-lowing.

Theorem 4 Under assumptions of Theorem 2, the value function of (ALM) has thedual representation

ϕ(c) = supy∈M∞

{〈c, y〉 − σC(y) − E

T∑

t=0

v∗t (yt )

}.

In general, the supremum in the above expression need not be attained; see Remark 7below. When C is a cone, the dual representation can be written

ϕ(c) = supy∈C∗

{〈c, y〉 − E

T∑

t=0

v∗t (yt )

},

where C∗ := {y ∈ M∞ | 〈c, y〉 ≤ 0 ∀c ∈ C ∩ M1} is the polar cone of C. This issimilar in form with the primal formulation

ϕ(c) = infd∈C

{E

T∑

t=0

vt (ct − dt )

}.

Example 5 Much of duality theory in optimal investment has studied the optimumvalue as a function of the initial endowment only; see e.g. Kramkov and Schacher-mayer [23] or Klein and Rogers [22]. The function V (c0) = ϕ(c0, 0, . . . , 0) gives the

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optimum value of (ALM) for an agent with initial capital −c0 and no future liabili-ties/endowments. Using (2), we can write this as

V (c0) = infd∈C(c0)

ET∑

t=0

vt (dt ), (6)

where the set C(c0) = {d ∈ M | (c0, 0, . . . , 0) − d ∈ C} consists of sequences ofinvestments needed to finance a riskless trading strategy when starting with initialcapital −c0. If ϕ is proper and lower semicontinuous (see Theorem 2), the biconjugatetheorem gives

V (c0) = supy∈M∞

{c0 y0 − ϕ∗(y)} = supy0∈R

{c0 y0 − U (y0)}

where

U (y0) = infz∈M∞{ϕ∗(z) | z0 = y0}.

If C is conical, we can use Lemma 3 to write U analogously to (6) as

U (y0) = infz∈Y(y0)

ET∑

t=0

v∗t (zt ),

where Y(y0) = {z ∈ C∗ | z0 = y0}. In general, there is no reason to believe that U islower semicontinuous nor that the infimum in its definition is attained. In some cases,however, it is possible to enlarge the set Y(y0) so that the function U becomes lowersemicontinuous and the infimum is attained; see [7, Section 4] for details.

In order to relate the above to more familiar duality results in optimal investment,we need an explicit expression for the support function σC . A cost process S is said tobe integrable if St (x, ·) is integrable for every x ∈ R

J and t = 0, . . . , T . Integrabilityimplies that domSt (·, ω) = R

J almost surely.4 A linear cost process St (x, ω) =st (ω) · x is integrable if and only if st are integrable. The following result, where N 1

denotes the space of integrable RJ -valued adapted processes and Et the conditional

expectation with respect to Ft , is from [30, Lemma A1]. Its proof is based on theclassical Fenchel–Moreau–Rockafellar duality theorem.

Lemma 6 If S is integrable, then

σC(y) = infz∈N 1

{T∑

t=0

E(yt St )∗(zt ) +

T −1∑

t=0

EσDt (Et�zt+1)

}

4 Integrability implies P(St (x, ·) ∈ R) = 1 for every x ∈ RJ so P(St (x, ·) ∈ R ∀x ∈ Q

J ) = 1, where Q

denotes the rational numbers. Since St (·, ω) are convex, this implies P(St (x, ω) ∈ R ∀x ∈ RJ ) = 1.

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T. Pennanen

for every y ∈ M∞+ while σC(y) = +∞ for y /∈ M∞+ . Moreover, the infimum isattained for every y ∈ M∞+ .

If S is sublinear and integrable (so that domSt = RJ and S∗

t is the indicator of itsdomain), we get (yt St )

∗(z) = δ(z | yt domS∗t ). If in addition, D is conical, we have

σDt = δD∗t

and

σC = δC∗ ,

where, by Lemma 6,

C∗ = {y ∈ M∞+ | ∃z ∈ N 1 : zt ∈ yt domS∗t , Et [�zt+1] ∈ D∗

t }.

The polar cone of C can also be written as5

C∗ = {y ∈ M∞+ | ∃s ∈ N : st ∈ domS∗t , Et [�(yt+1st+1)] ∈ D∗

t }.

In unconstrained linear models with St (x, ω) = st (ω) · x and Dt (ω) = RJ , we get

domS∗t = {st } and D∗

t (ω) = {0} so that

C∗ = {y ∈ M∞+ | ys is a martingale},

while in models with bid-ask spreads, we get domS∗t = [s−

t , s+t ] and

C∗ = {y ∈ M∞+ | ∃s ∈ N : s−t ≤ st ≤ s+

t , ys is a martingale}.

When short selling is prohibited, i.e. when Dt = RJ+, the condition Et [�(yt+1st+1)] ∈

D∗t means that ys is a supermartingale.

Remark 7 (Shadow prices) In unconstrained models with bid-ask spreads, the dualrepresentation of the value function can be written

ϕ(c) = supy∈M∞

{〈c, y〉 − E

T∑

t=0

v∗t (yt )

∣∣∣∣∣ ∃s ∈ N : s−t ≤ st ≤ s+

t , ys is a martingale

}.

If the supremum is attained, the corresponding s ∈ N has the property that the optimumvalue of (ALM) is not affected if trading costs are reduced by replacing S by the linearcost functions

St (x, ω) = st (ω) · x .

5 Here we use the fact that the integrability of S implies that every selector of domS∗t is integrable.

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Indeed, the corresponding value function ϕ satisfies

ϕ(c) ≥ ϕ(c)

≥ supy∈M∞

{〈c, y〉 − ϕ∗(y)}

= supy∈M∞

{〈c, y〉 − E

T∑

t=0

v∗t (yt )

∣∣∣∣∣ ys is a martingale

}.

where the first inequality holds because S ≥ S and the equality comes from Lemma 3.If ϕ is proper and lower semicontinuous and if the supremum in its dual representationis attained by a y ∈ M∞ and s, then the last supremum equals ϕ(c). In [7], priceprocesses s such that ϕ(c) = ϕ(c) are called shadow prices. If the disutility functionsvt are strictly increasing, then in the presence of a shadow price s, the optimal solutionx ∈ ND of (ALM), which exists under the assumptions of Theorem 2, must satisfythe complementarity conditions

�xt > 0 �⇒ st = s+t and �xt < 0 �⇒ st = s−

t

since otherwise x would achieve strictly lower trading costs and thus, a strictly lowerobjective value in the model with the linear cost functions St . Section 3 of [7], givesan example where shadow prices do not exist and thus, the supremum in the dualrepresentation of ϕ is not attained.

In the presence of a numeraire (see Example 1), the elements of C∗ can be expressedin terms of probability measures.

Corollary 8 (Numeraire and martingale measures) Assume that S is integrable andthat

St (x, ω) = x0 + St (x, ω) and Dt (ω) = R × Dt (ω),

where S is sublinear and D is conical. Then6

C∗ = pos

{y ∈ M∞+ | ∃Q ∈ P : yt = Et

d Q

d P

}

where P = {Q � P | ∃s ∈ N : st ∈ dom S∗t , E Q

t �st+1 ∈ D∗t Q-a.s.} and N

denotes the set of adapted RJ\{0}-valued processes.

Proof We get

domSt = {1} × dom S∗t and D∗

t = {0} × D∗t

6 For a subset C of a vector space, posC := {αx | α ≥ 0, x ∈ C}.

123

T. Pennanen

so, by Lemma 6, y ∈ C∗ iff y is a nonnegative martingale and Et [�(yt+1st+1)] ∈ D∗t

for some (Ft )Tt=0-adapted selector s of dom S∗ such that ys is integrable. Clearly,

C∗ = pos{y ∈ C∗ | EyT = 1}. If y ∈ C∗ with EyT = 1, then yT is the density of aprobability measure Q � P and, by [12, Proposition A.12],

E Qt [�st+1] = Et [yT �st+1]

Et yT= Et [�(yt+1st+1)]

Et yTQ-a.s.

Since D∗t is a cone, this implies E Q

t [�st+1] ∈ D∗t Q-almost surely. Thus, {y ∈

C∗ | EyT = 1} ⊆ {y ∈ M∞+ | ∃Q ∈ P : yt = Etd Qd P }. Conversely, if yt =

Et [d Q/d P] for some Q ∈ P , we get similarly that

Et [yT �st+1] ∈ D∗t Q-a.s..

Since Et [yT �st+1] = 0 P-almost surely on any A ∈ Ft with Q(A) = 0 and since0 ∈ D∗

t , we get

Et [yT �st+1] ∈ D∗t P-a.s.

and thus, y ∈ C∗. We thus have

{y ∈ C∗ | EyT = 1

} ={

y ∈ M∞+ | ∃Q ∈ P : yt = Etd Q

d P

}

which completes the proof. ��In the classical linear model with St (x, ω) = st (ω) · x and Dt = R

J , we havedomS∗

t = {st } and D∗t = {0} so the set P in Corollary 8 becomes the set of absolutely

continuous martingale measures. In unconstrained models with bid-ask spreads, Pis the set of absolutely continuous probability measures Q such that there exists aQ-martingale s with s−

t ≤ st ≤ s+t for all t .

Example 9 In the setting of Corollary 8, the dual representation of ϕ in Theorem 4can be written as

ϕ(c) = supy∈C∗

ET∑

t=0

[ct yt − v∗t (yt )]

= supλ≥0

supQ∈P

ET∑

t=0

{ctλEt

d Q

d P−

T∑

t=0

v∗t

(λEt

d Q

d P

)}

= supλ≥0

supQ∈P

{λE Q

T∑

t=0

ct − ET∑

t=0

v∗t

(λEt

d Q

d P

)}.

This is an illiquid discrete-time version of the dual problem of the optimal consumptionproblem from Karatzas and Zitkovic [21] which extends the duality framework of

123


Kramkov and Schachermayer [23] by allowing consumption of wealth over time.When vt = δR− for t < T , we obtain a discrete-time version with illiquidity effectsof the duality results in Owen and Zitkovic [29] and Biagini, Frittelli and Graselli [3].In the exponential case

vT (c) = 1

α(eαc − 1),

we get v∗T (y) = (y ln y − y + 1)/α and the supremum over λ is attained at

λ = exp

(E

(α

T∑

t=0

ct yt − yT ln yT

)).

This gives

ϕ(c) = supQ∈P

{1

αexp

[αE Q

T∑

t=0

ct − H(Q|P)

]− 1

α

}

= 1

αexp

[supQ∈P

{αE Q

T∑

t=0

ct − H(Q|P)

}]− 1

α.

In the linear case St (x) = st · x , this reduces to a discrete-time version of the dualityframework of [9]; see also [20] and [1].

4 Conclusions

This paper studied optimal investment in the general conjugate duality frameworkof convex analysis. This has the benefit of allowing for various generalizations to theclassical market models based on the theory of stochastic integration. In particular, theintroduction of portfolio constraints and nonlinear illiquidity effects poses no particularproblems compared to the classical model of perfectly liquid markets. One could alsoinclude dividend payments as proposed in [4]. Separation of dividend payments from“total returns” is important in the presence of transaction costs.

The optimization problem studied in this paper has important applications inaccounting and in the valuation of swap contracts and other financial products withmultiple payout dates. This has been studied in [34]. The related duality theory willbe developed elsewhere.

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Convex duality in optimal investment under illiquidity